Modular Properties of 3D Higher Spin Theory

Understand

In the three-dimensional sl(N) Chern-Simons higher-spin theory, we prove that the conical surplus and the black hole solution are related by the S-transformation of the modulus of the boundary torus.

• Then applying the modular group on a given conical surplus solution, we generate a 'SL(2,Z)' family of smooth constant solutions.
• We then show how these solutions are mapped into one another by coordinate transformations that act non-trivially on the homology of the boundary torus.
• After deriving a thermodynamics that applies to all the solutions in the 'SL(2,Z)' family, we compute their entropies and free energies, and determine how the latter transform under the modular transformations.